Noether's Theorem for a Fixed Region

TL;DR

This paper presents an elementary proof of Noether's first Theorem emphasizing that global quasi-symmetry needs only to hold for a fixed region, and discusses conditions for gauging such symmetries.

Contribution

It offers a simplified proof of Noether's Theorem highlighting the fixed region aspect and provides criteria for gauging global quasi-symmetries.

Findings

Global quasi-symmetry suffices for Noether's Theorem when holding on one fixed region

Conditions for gauging a global quasi-symmetry are established

Elementary proof approach simplifies understanding of Noether's Theorem

Abstract

We give an elementary proof of Noether's first Theorem while stressing the magical fact that the global quasi-symmetry only needs to hold for one fixed integration region. We provide sufficient conditions for gauging a global quasi-symmetry.